Noethericity and index of a characteristic bisingular integral operator with shifts. (English. Russian original) Zbl 1476.47012

Efimov, Sergeĭ V.

Sib. Math. J. 60, No. 4, 585-591 (2019); translation from Sib. Mat. Zh. 60, No. 4, 751-759 (2019).

Summary: We consider a characteristic bisingular operator with rather arbitrary shifts that decompose into one-dimensional components. We reduce the problem about the Noethericity and index to that about an operator without shifts. The results obtained are straightforwardly applicable to the two-dimensional boundary-value problem with shifts which is a natural generalization of the Haseman and Carleman problems.

MSC:

47A53	(Semi-) Fredholm operators; index theories
47A05	General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47G10	Integral operators

Keywords:

Noethericity; index; bisingular operator; shift

References:

[1]	Efimov S. V., “Noethericity and index of characteristic bisingular operator with shift,” Vladikavkazsk. Mat. Zh., vol. 16, no. 2, 46-48 (2014). · Zbl 1330.47022
[2]	Sazonov L. I., “A bisingular equation with translation in the space Lp,” Math. Notes, vol. 13, no. 3, 235-239 (1973). · Zbl 0265.45006
[3]	Pilidi V. S. and Stefanidi E. N., “On an algebra of bisingular operators with shift,” Rostov-on-Don, 1981. 26 pp. Submitted to VINITI, no. 3036. · Zbl 0499.47031
[4]	Pilidi V. S. and Stefanidi E. N., “On an algebra of bisingular operators with shift,” Izv. Vyssh. Uchebn. Zaved. Mat., no. 9, 80-81 (1981). · Zbl 0483.47035
[5]	Efimov S. V., “Bisingular operators with irreducible involutive shift,” Russian Math. (Iz. VUZ), no. 2, 29-36 (1992).
[6]	Efimov, S. V., On effectively verifiable conditions of the Noethericity of some bisingular operators with shift, 75-78 (1997), Rostov-on-Don
[7]	Efimov, S. V., Index of some bisingular integral operators with shift, 61-66 (1998), Rostov-on-Don
[8]	Efimov, S. V., Index of some bisingular operators with irreducible shift, 88-94 (2001), Rostov-on-Don
[9]	Efimov S. V., “Calculation of the index of some bisingular operators with irreducible involutive shift,” Izv. Vyssh. Uchebn. Zaved. Severo-Kavkaz. Reg. Estestv. Nauk, no. 9, 7-14 (2004). · Zbl 1082.58020
[10]	Efimov S. V., “On the index of some bisingular integral operators with shift,” Vestnik Don State Technical University, vol. 4, no. 3, 290-295 (2004).
[11]	Efimov S. V., “Calculation of the index of some bisingular operators with shift by the homotopy method,” Vestnik Don State Technical University, vol. 10, no. 1, 22-27 (2010).
[12]	Pilidi V. S., “On a bisingular equation in the space Lp,” in: Mat. Issled., Shtiintsa, Kishinev, 1972, vol. 7, no. 3, 167-175.
[13]	Pilidi V. S., “Index computation for a bisingular operator,” Funct. Anal. Appl., vol. 7, no. 4, 337-338 (1973). · Zbl 0295.47053
[14]	Gokhberg I. Ts. and Krupnik N. Ya., An Introduction to the Theory of One-Dimensional Singular Integral Operators [Russian], Shtiintsa, Kishinev (1973).
[15]	Litvinchuk G. S., Boundary Value Problems and Singular Integral Equations with Shift [Russian], Nauka, Moscow (1977). · Zbl 0462.30029
[16]	Simonenko I. B., “Some general questions in the theory of the Riemann boundary problem,” Math. USSR-Izv., vol. 2, no. 5, 1091-1099 (1968). · Zbl 0186.13601
[17]	Lavrentiev M. A. and Shabat B. V., Methods of the Theory of Functions of Complex Variables [Russian], Nauka, Moscow (1987).
[18]	Goluzin G. M., Geometric Theory of Functions of a Complex Variable, Amer. Math. Soc., Providence (1969). · Zbl 0183.07502